Composting/Post processing animation
Going through each part of the question, one by one:


Post-processing and compositing are pretty much the same thing. Post processing is generally more color oriented, and compositing is a bit more VFX oriented (keying etc. ). But for all intents and purposes in blender, they are the same thing, and done at the same time, in the same place. 
Yes, you can key frame nearly every single property value in the compositor, the key is just to choose the right ones when working with colors, or odd things can happen...
Yes, you'll do the compositing/post-pro last, You can do it all in blender, or use different software, like Adobe After effects, or the now 'free'* Fusion 7. For the color stuff (Post-Pro), you could use After Effects, Adobe Speed Grade, or DaVinci Resolve. Most of this software isn't cheap, which is why I recomend doing it in blender, which for the large part, has a fully capable feature set.


*Non commercial, with some non-key features removed, the full version costs about $1000